| An introduction to diagrammatic algebra and categorified quantum sl(2) - Aaron D. Lauda This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescalling isomorphisms for categorified quantum sl2, a proof that cyclotomic quotients of the nilHecke algebra are isomorphic to matrix rings over the coh... Downloads: 3 | |

| Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties - Aaron D. Lauda A 2-category was introduced in arXiv:0803.3652 [math.QA] that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N arepresentation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2). Downloads: 2 | |

| Frobenius algebras and planar open string topological field theories - Aaron D. Lauda Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a n... Downloads: 4 | |

| Frobenius algebras and ambidextrous adjunctions - Aaron D. Lauda In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction i... Downloads: 5 | |

| Remarks on 2-Groups - Aaron D. Lauda A 2-group is a `categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G x G -> G has been replaced by a functor. A number of precise definitions of this notion have already been explored, but a full treatment of their relationships is difficult to extract from the literature. Here we describe the relation between two of the most important versions of this notion, which we call `weak' and `coherent' 2-groups... Downloads: 4 | |

| A categorification of quantum sl(2) - Aaron D. Lauda We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our grap... Downloads: 3 | |

| Categorified quantum sl(2) is an inverse limit of flag 2-categories - Anna Beliakova We prove that categorified quantum sl(2) is an inverse limit of Flag 2-categories defined using cohomology rings of iterated flag varieties. This inverse limit is an instance of a 2-limit in a bicategory giving rise to a universal property that characterizes the categorification of quantum sl(2) uniquely up to equivalence. As an application we characterize all bimodule homomorphisms in the Flag 2-category and prove that the categorified quantum Casimir of sl(2) acts appropriately on these 2-repr... Downloads: 4 | |

| Open-closed TQFTs extend Khovanov homology from links to tangles - Aaron D. Lauda We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suitable algebra A such that there is one action of A or A^op for every boundary point of the tangle... Downloads: 6 | |

| Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras - Aaron D. Lauda We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras... Downloads: 7 | |

| Crystals from categorified quantum groups - Aaron D. Lauda We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendiec... Downloads: 5 | |

| A diagrammatic approach to categorification of quantum groups I - Mikhail Khovanov To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph. Downloads: 3 | |

| A diagrammatic approach to categorification of quantum groups III - Mikhail Khovanov We categorify the idempotented form of quantum sl(n). Downloads: 26 | |

| Implicit structure in 2-representations of quantum groups - Sabin Cautis Given a strong 2-representation of a Kac-Moody Lie algebra (in the sense of Rouquier) we show how to extend it to a 2-representation of categorified quantum groups (in the sense of Khovanov-Lauda). This involves checking certain extra 2-relations which are explicit in the definition by Khovanov-Lauda and, as it turns out, implicit in Rouquier's definition. Some applications are also discussed. Downloads: 13 | |

| State sum construction of two-dimensional open-closed Topological Quantum Field Theories - Aaron D. Lauda We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma--Hosono--Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is ba... Downloads: 18 | |

| A diagrammatic approach to categorification of quantum groups II - Mikhail Khovanov We categorify one-half of the quantum group associated to an arbitrary Cartan datum. Downloads: 43 | |

| A categorification of the Casimir of quantum sl(2) - Anna Beliakova We categorify the Casimir element of the idempotented form of quantum sl(2). Downloads: 4 | |

| Higher-Dimensional Algebra V: 2-Groups - John C. Baez A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y t... Downloads: 6 | |

| Nilpotency in type A cyclotomic quotients - Alexander E. Hoffnung We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree of cyclotomic quotients of rings that categorify one-half of quantum sl(k). Downloads: 10 | |

| Oddification of the cohomology of type A Springer varieties - Aaron D. Lauda We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions... Downloads: 18 | |

| The odd nilHecke algebra and its diagrammatics - Alexander P. Ellis We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians... Downloads: 5 | |

| Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) - Aaron D. Lauda We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Utilizing Cautis-Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones-Wenzl projectors and their sl(3) analogs purely from the h... Downloads: 11 | |

| Extended graphical calculus for categorified quantum sl(2) - Mikhail Khovanov A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection w... Downloads: 7 | |